arXiv subjects

Robert A. Van Gorder

Publications and source records attributed to Robert A. Van Gorder.

20 records · Page 2Linked to original sources

arXiv2012

Unstaggered-staggered solitons in two-component discrete nonlinear Schrödinger lattices

We present stable bright solitons built of coupled unstaggered and staggered components in a symmetric system of two discrete nonlinear Schrödinger (DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity, coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. The results are obtained in an analytical form, using the variational and Thomas-Fermi approximations (VA and TFA), and the generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the stability. The analytical predictions are verified against numerical results. Almost all the symbiotic solitons are predicted by the VA quite accurately, and are stable. Close to a boundary of the existence region of the solitons (which may feature several connected branches), there are broad solitons which are not well approximated by the VA, and are unstable.

nlin.PS↗

arXiv2011

Resonant solitons from the $3\times 3$ operator

Resonant solitons of the $3\times 3$ operator are studied. The scattering data of this operator contains four transmission coefficients, two in each half complex $ζ$-plane, where $ζ$ is the spectral parameter. For anti-hermitian symmetry of the potential, the two transmission coefficients in the lower half plane (LHP) become equal to the complex conjugates of those in the upper half plane (UHP). The bound state scattering data for this operator consists in part of the zeros of these two transmission coefficients. Of particular interest is that class of soliton solutions when the two transmission coefficients have exactly equal eigenvalues, which gives rist to "resonant solitons". They arise from a bifurcation which is caused by the algebraic structure of the $3\times 3$ scattering matrix. We detail the asymptotics of this solution, showing that the latter contains the well known parametric interactions of "up-conversion" and "down-conversion". Lastly, we explain how this equality of eigenvalues in different transmission coefficients can be seen to be a nonlinear resonance condition.

nlin.SI↗